James has been in sparkling form since joining Everton from Real Madrid on a free transfer last month, scoring three goals in his first four Premier League outings.
Clinching a victory in his 100th match at Roland Garros, the record 12-time champion secured a place in the last four for 13th time in the clay major, ending an impressive run from Sinner who had upset 11th seed David Goffin and 6th seed Alexander Zverev in his previous matches.
Eibar midfielder Pape Diop was sent off for the second time this season with 18 minutes remaining, but Rodrigues' later header gave them three vital points in injury time. The visitors also had to thank goalkeeper Marco Dmitrovic for an excellent penalty save with the score at 1-1.
FC Barcelona have had a promising start to the campaign under new coach Ronald Koeman, scoring seven goals in their opening two matches, with the midfield pairing of Sergio Busquets and Frenkie de Jong giving a solid base for the attacking quarter of Ansu Fati, Leo Messi, Philippe Coutinho and Antoine Griezmann.
The Buenos Aires giants confirmed the move via a brief statement on social media, in which they wished Marcone "many successes in this new stage".
"I'm very happy. It was a great match, I didn't make any mistakes. Casper is a really great player, very attractive to play, I think there were many good rallies, it was a good match from both of us," said the recently-crowned US Open champion.
James, 29, has also impressed in his opening three matches for Everton in the Premier League following his summer move from Real Madrid.
The river-crossing competition was one of the activities of the 2020 Liuzhou International Water Carnival.
The 19-year-old Pole needed just 68 minutes to avenge her exit to the 2018 champion last year in Paris at the same stage.
The Spaniard, who is seeking his 13th title at Roland Garros, defeated Sinner 7-6 (7-4), 6-4, 6-1 to set up a rematch with Diego Schwartzman on Friday.
Esteban Burgos scored from the penalty spot for Eibar after 28 minutes only for Toni Villa to equalize for the home team eight minutes later.